An asymptotic expansion for the null distribution of the efficient score statistic

An asymptotic expansion for the null distribution of the efficient score statistic for testing a composite hypothesis in the presence of nuisance parameters is derived, and an interpretation of the terms occurring in the expansion is given. The use of the expansion to modify the percentage points of the large-sample reference x2 distribution is discussed. The first three moments of the null distribution are obtained and are used as a check on the accuracy of the algebra via a comparison with the first three moments of the index of dispersion test for homogeneity of Poisson parameters.